Similar to standard spectral techniques, the objective of \(p\) -spectral clustering is to approximate multiple mutually orthogonal eigenvectors of a nonlinear generalization of the graph Laplacian matrix, which will subsequently be considered as the \(p\) -spectral coordinates of the nodes of the graph. The orthogonality constraint combined with the fact that the final partitioning step takes place in the reduced space of \(p\) -spectral coordinates suggests that recasting the problem as an optimization procedure over a subspace that adheres by definition to these conditions, i.e., the Grassmann manifold, would be beneficial. The Grassmann manifold is defined as the set of all \(k\) -dimensional subspaces of \(\mathbb {R}^n\) , where \(k,n \in \mathbb {Z}^{+}, k \le n\) , and it is denoted here by \(\mathcal {Gr}(k,n)\) . Its Riemannian structure, and the development of robust optimization algorithms in this manifold, have been extensively researched [1]}, [2]}, [3]}.
